You will need to know multivariable calculus, linear algebra, and ordinary differential equations.
That's for "basic" QM (which is what you asked for), so I am leaving out partial differential equations and group theory.
Why not PDEs? Well, while the Schrodinger equation is a PDE, you will be separating it into 4 ODEs (3 space+1 time), and all your ODE methods can be brought to bear.
Why not group theory? The only algebra you'll need for a first course in QM is on matrix manipulations, vector spaces, and linear transformations. You certainly will not have to know about SU(N), generators of groups, etc. The only thing from group theory that will come up is the idea of a nonabelian group, but this will all be self-contained in commutation relations.
For an advanced undergraduate course, you will need to supplement your mathematics with both PDEs and groups.
Still depends on the course material and how much math the teacher knows and is willing to include in the course.For example,my course assumed basic notions of finite dimensional vector spaces (linear algebra),the elementary notions of topology (set,open set,closed set),partial differential equations (for the exercise part),a bit about the functional "delta-Dirac".And that's just about it,because my teacher knew all the math (functional analysis) the understanding of the principles required and,moreover,was willing to teach us.
PS.Integration of functions with maximum 3 variables and special functions (orthogonal polynomials,Bessel functions).
The Superstring Theory website has a three-tiered list of mathematical areas one would need to know for: (1) undergraduate physics; (2) graduate students in theoretical physics; and (3) "hot topics" in string theory.
As a non-physicist myself, I'm unable to say how well the same courses would map onto what is needed for quantum mechanics. However, the section on mathematical background for graduate study in theoretical physics seems like it could be relevant.